Sequential Posted Price Mechanisms with Correlated Valuations Marek Adamczyk marek.adamczyk@tum.de Technische Universität München Allan Borodin bor@cs.toronto.edu University of Toronto Diodato Ferraioli dferraioli@unisa.it University of Salerno Bart de Keijzer keijzer@cwi.nl Centrum Wiskunde & Informatica (CWI), Amsterdam Stefano Leonardi leonardi@dis.uniroma.it Sapienza University of Rome Abstract We study the revenue performance of sequential posted price mechanisms and some natural extensions, for a general setting where the valuations of the buyers are drawn from a correlated distribution. Sequential posted price mechanisms are conceptually simple mechanisms that work by proposing a take-it-or-leave-it offer to each buyer. We apply sequential posted price mechanisms to single-parameter multi-unit settings in which each buyer demands only one item and the mechanism can assign the service to at most k of the buyers. For standard sequential posted price mechanisms, we prove that with the valuation distribution having finite support, no sequential posted price mechanism can extract a constant fraction of the optimal expected revenue, even with unlimited supply. We extend this result to the the case of a continuous valuation distribution when various standard assumptions hold simultaneously (i.e., everywhere-supported, continuous, symmetric, and normalized (conditional) distributions that satisfy regularity, the MHR condition, and affiliation). In fact, it turns out that the best fraction of the optimal revenue that is extractable by a sequential posted price mechanism is proportional to ratio of the highest and lowest possible valuation. We prove that a simple generalization of these mechanisms allows to achieve a better revenue performance: if the sequential posted price mechanism has for each buyer the option of either proposing an offer or asking the buyer for its valuation, then a Ω(/ max{, d}) fraction of the optimal revenue can be extracted, where d denotes the degree of dependence of the valuations, ranging from complete independence (d = 0) to arbitrary dependence (d = n ). In order to prove this result we generalize the sequential posted price mechanisms further, such that the mechanism has the ability to make a take-it-or-leave-it offer to the i th buyer that depends on the valuations of all buyers except i s. We prove that a constant fraction (2 e)/4 0.088 of the optimal revenue can be always extracted by this generalized mechanism. Introduction A large body of literature in the field of mechanism design focuses on the design of auctions that are optimal with respect to some given objective function, such as maximizing the social welfare or the auctioneer s revenue. This literature mainly considered direct revelation mechanisms, in which each buyer submits a bid that represents his valuation for getting the service, and the mechanism determines the winners and how much they are forced to pay. The reason for this is the revelation principle (see, e.g., Börgers [205]), which implies that one may resort to studying only direct revelation mechanisms for many purposes, such as maximizing the social welfare or the auctioneer s revenue. Some of the most celebrated mechanisms follow this approach, such as the Vickrey-Clark-Groves Mechanism [Vickrey, 96, Clarke, 97, Groves, 973] and the Myerson Mechanism [Myerson, 98]. A natural assumption behind these mechanisms is that buyers will submit truthfully whenever the utility they take with the truthful bid is at least as high as the utility they may take with a different bid. However, it has often been acknowledged that such an assumption may be too strong in a real world setting. In particular, Sandholm and Gilpin [2004] highlight that this assumption frequently fails because of: (i) a buyer s unwillingness to fully specify their values, (ii) a buyer s unwillingness to participate in ill understood, complex, unintuitive auction mechanisms, and 3) irrationality of a buyer, which leads him to underbid even when it is known that there is nothing to be gained from this behavior. The failure of direct revelation mechanisms is also confirmed by experimental studies (see, e.g., Kagel et al. [987]).
This has recently motivated the research about auction mechanisms that are conceptually simple. Among these, the class of sequential posted price mechanisms [Chawla et al., 200] is particularly attractive. First studied by Sandholm and Gilpin [2004] (and called take-it-or-leave-it mechanisms ), these mechanisms work by iteratively selecting a buyer that has not been selected previously, and offering him a price. The buyer may then accept or reject that price. When the buyer accepts, he is allocated the service. Otherwise, the mechanism does not allocate the service to the buyer. In the sequential posted-price mechanism we allow both the choice of buyer and the price offered to that buyer to depend on the decisions of the previously selected buyers (and the prior knowledge about the buyers valuations). Also, randomization in the choice of the buyer and in the charged price is allowed. Sequential posted price mechanisms are thus conceptually simple and buyers do not have to reveal their valuations for the service. Moreover, they are individually rational, i.e., informally, participation in such an auction is never harmful to the buyer. Finally, they possess a trivial dominant strategy: indeed, it is always best for a buyer to respond sincerely to any offer that the mechanism makes him. (equivalently, following the definition of Li [205], we can say that these mechanisms are obviously strategy proof ). Sequential posted price mechanisms have been mainly studied for the setting where the valuations of the buyers are each drawn independently from publicly known buyer-specific distributions, called the independent values setting. In this paper, we study a much more general setting, and assume that the entire vector of valuations is drawn from one publicly known distribution, which allows for arbitrarily complex dependencies among the valuations of the buyers. This setting is commonly known as the correlated values setting. Our goal is to investigate questions related to the existence of sequential posted price mechanisms that achieve a high revenue. That is, we quantify the quality of a mechanism by comparing its expected revenue to that of the optimal mechanism, defined as the mechanism that achieves the highest expected revenue among all dominant strategy incentive compatible and ex-post individually rational mechanisms. (In Section 2, we explain and motivate this optimality notion further.) We assume a standard Bayesian, transferable, quasi-linear utility model and we study a unit demand, single parameter, multi-unit setting: there is one service (or type of item) being provided by the auctioneer, any buyer is interested in receiving the service (or item) once, and the valuation of each buyer consists of a single number that reflects to what extent a buyer would profit from receiving the service provided by the auctioneer. The auctioneer can charge a price to a bidder, so that the utility of a bidder is his valuation (in case he gets the service), minus the charged price. We focus in this paper on the k-limited supply setting, where service can be provided to at most k of the buyers. This is an important setting because it is a natural constraint in many realistic scenarios, and it contains two fundamental special cases: the unit supply setting (where k = ), and the unlimited supply setting where k = n. Related work There has been recent substantial work on the subject of revenue performance for simple mechanisms [Hartline and Roughgarden, 2009, Hart and Nisan, 202, Babaioff et al., 204, Rubinstein and Weinberg, 205, Devanur et al., 205]. In particular, Babaioff et al. [204] highlight the importance of understanding what is the relative strength of simple versus complex mechanisms with regard to revenue maximization. As described above, sequential posted price mechanisms are an example of such a simple class of mechanisms. Sandholm and Gilpin [2004] have been the first ones to study sequential posted price mechanisms. They give experimental results for the case in which values are independently drawn from the uniform distribution in [0, ]. Moreover, they consider the case where multiple offers can be made to a bidder, and study the equilibria that arise from this. Blumrosen and Holenstein [2008] compare fixed price (called symmetric auctions), sequential posted price (called discriminatory auctions) and the optimal mechanism for valuations drawn from a wide class of i.i.d. distributions. Babaioff et al. [202] consider prior-independent posted price mechanisms with k-limited supply for the setting where the only information known about the valuation distribution is that all valuations are independently drawn from the same distribution with support [0, ]. Posted-price mechanisms have also been previously studied in [Kleinberg and Leighton, 2003, Blum and Hartline, 2005, Blum et al., 2004], albeit for a non-bayesian, on-line setting. In a recent work Feldman et al. [205a] study on-line posted price mechanisms for combinatorial auctions when valuations are independently drawn. The works of Chawla et al. [200] and Gupta and Nagarajan [203] are closest to our present work, although they only consider sequential posted price mechanisms in the independent values setting. In particular, Chawla et al. [200] prove that such mechanisms can extract a constant factor of the optimal revenue for single and multiple parameter settings under various constraints on the allocations. They also consider order-oblivious (i.e., on-line ) sequential posted price mechanisms in which the order of the buyers is fixed and adversarially determined. They use orderoblivious mechanisms in order to establish some results for the more general multi-parameter case. Yan [20] builds on this work and strengthens some of the results of Chawla et al. [200]. Moreover, Kleinberg and Weinberg [202] 2
and Feldman et al. [205b] prove results that imply a strengthening of some of the results of Chawla et al. [200]. Gupta and Nagarajan [203] introduce a more abstract stochastic probing problem that includes Bayesian sequential posted price mechanisms as well as the stochastic matching problem introduced by Chen et al. [2009]. Their approximation bounds were later improved by Adamczyk et al. [204] who in particular matched the approximation of Chawla et al. [200] for the case where there is a single matroid feasibility constraint. All previous work only consider the independent setting. In this work, we instead focus on the correlated setting. The lookahead mechanism of Ronen [200] is a fundamental reference for the correlated setting and resembles our blind offer mechanism but is different in substantial ways, as we will discuss below. Cremer and McLean [988] made a fundamental contribution to auction theory in the correlated value setting, by characterizing exactly for which valuation distributions it is possible to extract the full optimal social welfare as revenue. They do this for the ex-post IC, interim IR mechanisms and for the dominant strategy IC, interim IR mechanisms. Segal [2003] gives a characterization of optimal ex-post incentive compatible and ex-post individually rational optimal mechanisms. Roughgarden and Talgam-Cohen [203] study optimal mechanism design in the even more general interdependent setting. They show how to extend the Myerson mechanism to this setting for various assumptions on the valuation distribution. There is now a substantial literature [Dobzinski et al., 20, Roughgarden and Talgam-Cohen, 203, Chawla et al., 204], that develops mechanisms with good approximation guarantees for revenue maximization in the correlated setting. These mechanisms build on the lookahead mechanism of Ronen [200] and thus they also differ from the mechanisms proposed in this work. Contributions and outline We define some preliminaries and notation in Section 2. In Section 3. we first present a simple sequence of instances. It shows that for unrestricted correlated distributions, the expected revenue of the best sequential posted price mechanism does not approximate within any constant factor the expected revenue of the optimal dominant strategy incentive compatible and ex-post individually rational mechanism. This holds for any value of k (i.e, the size of the supply). We extend this impossibility result by proving that a constant approximation is impossible to achieve even when we assume that the valuation distribution is continuous and satisfies all of the following conditions simultaneously: the valuation distribution is supported everywhere, is entirely symmetric, satisfies regularity, satisfies the monotone hazard rate condition, satisfies affiliation, all the induced marginal distributions have finite expectation, and all the conditional marginal distributions are non-zero everywhere. The maximum revenue that a sequential posted price mechanism can generate on our examples is shown to be characterized by the logarithm of the ratio between the highest and lowest valuations in the support of the distribution. We show in Section 3.2 that this approximation ratio is essentially tight. Given these negative results, we consider a generalization of sequential posted price mechanisms that are more suitable for settings with limited dependence among the buyers valuations: enhanced sequential posted price mechanisms. An enhanced sequential posted price mechanism works by iteratively selecting a buyer that has not been selected previously. The auctioneer can either offer the selected buyer a price or ask him to report his valuation. As in sequential posted price mechanisms, if the buyer is offered a price, then he may accept or reject that price. When the buyer accepts, he is allocated the service. Otherwise, the mechanism does not allocate the service to the buyer. If instead, the buyer is asked to report his valuation, then the mechanism does not allocate him the service. Note that the enhanced sequential posted price mechanism requires that some fraction of buyers reveal their valuation truthfully. Thus, the original property that the bidders not have to reveal their preferences is partially sacrificed, in return for a more powerful class of mechanisms and (as we will see) a better revenue performance. For practical implementation, such mechanisms can be slightly adjusted by providing a bidder with a small monetary reward in case he is asked to reveal his valuation. For the enhanced sequential posted price mechanisms, we prove that again there are instances in which the revenue is not within a constant fraction of the optimal revenue. However, we show that this class of mechanisms can extract a fraction Θ(/n) of the optimal revenue, i.e., a fraction that is independent of the valuation distribution. This result seems to suggest that to achieve a constant approximation of the optimal revenue it is necessary to collect all the bids truthfully. Consistent with this hypothesis, we prove that a constant fraction of the optimal revenue can be extracted by dominant strategy IC blind offer mechanisms: these mechanisms inherit all the limitations of sequential posted price mechanisms (i.e., buyers are considered sequentially in an order independent of their submitted bids; it is possible to offer a price to a buyer only when selected; and the buyer gets the service only if it accepts the We also note that in some realistic scenarios, the valuation of some buyers may be known a priori to the auctioneer (through, for example, the repetition of auctions or accounting and profiling operations). In such settings, though it is not necessary to query such a buyer for his valuation, our results for enhanced sequential posted price mechanisms are also relevant. 3
offered price), except that the price offered to a bidder i may now depend on the bids submitted by all buyers other than i. This generalization sacrifices entirely the property that buyers valuations do not need to be revealed, and blind offer mechanisms are thus necessarily direct revelation mechanisms. However, this comes with the reward of a revenue that is only a constant factor away from optimal. Also, blind offer mechanisms preserve the conceptual simplicity of sequential posted price mechanisms, and are easy to grasp for the buyers participating in the auction: in particular, buyers have a conceptually simple and practical strategy, to accept the price if and only if it is not above their valuation, regardless of how the prices are computed. Unfortunately it turns out that blind offer mechanisms are not inherently truthful, although this is only for a subtle reason: while it is true that a buyer cannot influence the price he is offered nor the decision of the mechanism to pick him, he can still be incentivized to misreport his valuation in order to influence the probability that the supply has not run out before the buyer is picked. However, this is not a big obstruction, as we will show that there is a straightforward way to turn a blind offer mechanism into an incentive compatible one, with only a small loss in approximation factor. Even though blind offer mechanisms sacrifice some simplicity (and practicality), it is still interesting that a mechanism that allocates items to buyers on-line (i.e., it works regardless of the order in which buyers are served) is able to achieve a constant approximation of the optimal revenue even with correlated valuations. On the contrary, all previously known mechanisms (as, e.g., Ronen [200] and Chawla et al. [200])) that achieve a constant approximation with correlated valuations allocate the items to the buyers that maximize the profit. Thus they do not work for every order, but only for very specific ones. Moreover, our result for blind offer mechanisms has a constructive purpose: it provides the intermediate step in the tentative of establishing revenue approximation bounds for other mechanisms. We will show how blind offer mechanisms serve to this purpose in Section 4 for the construction of good enhanced sequential posted price mechanisms. We highlight that our positive results do not make any assumptions on the marginal valuation distributions of the buyers nor the type of correlation among the buyers. An exception is Section 4, where we consider the case in which the degree of dependence among the buyers is limited. In particular, we introduce the notion of d-dimensionally dependent distributions. This notion informally requires that for each buyer i there is a set S i of d other buyers such that the distribution of i s valuation when conditioning on the vector of other buyers valuations can likewise be obtained by only conditioning on the valuations of S i. Thus, this notion induces a hierarchy of n classes of valuation distributions with increasing degrees of dependence among the buyers: for d = 0 the buyers have independent valuations, while the other extreme d = n implies that the valuations may be dependent in arbitrarily complex ways. Note that d-dimensional dependence does not require that the marginal valuation distributions of the buyers themselves satisfy any particular property, and neither does it require anything from the type of correlation or dependence that may exist among the buyers. This stands in contrast with common assumptions such as symmetry, affiliation, the monotone-hazard rate assumption, and regularity, that are often encountered in the auction theory and mechanism design literature. Our main positive result for enhanced sequential posted price mechanisms then states that when the valuation distribution is d-dimensionally dependent, there exists an enhanced sequential posted price mechanism that extracts an Ω(/d) fraction of the optimal revenue. The proof of this result consists of three key ingredients: An upper bound on the optimal ex-post IC, ex-post IR revenue in terms of the solution of a linear program. This upper bound has the form of a relatively simple expression that is important for the definition and analysis of a blind offer mechanism that we define subsequently. This part of the proof generalizes a linear programming characterization introduced by Gupta and Nagarajan [203] for the independent distribution setting. A proof for the fact that incentive compatible blind offer mechanisms are powerful enough to extract a constant fraction of the optimal revenue of any instance. (Note that this is in itself the main result that we prove for blind offer mechanisms.) This makes crucial use of the linear program mentioned above. A conversion lemma showing that blind offer mechanisms can be turned into enhanced sequential posted price mechanisms while maintaining a fraction Ω(/d) of the revenue of the blind offer mechanism 2. While our focus is on proving the (non-)existence of simple mechanisms that perform well in terms of revenue, we note the following about the computational complexity of our mechanisms: all the mechanisms that we use in our 2 The sequential posted price mechanism constructed in this lemma depends on a parameter q which can be adjusted to trade off the amount of valuation elicitation against the hidden constant in the Ω(/d) expression. 4
positive results run in polynomial time when the valuation distribution is given as a description of the valuation vectors together with their probability mass 3. Additionally, we note that all of our negative results hold for randomized mechanisms. On the other hand, our positive results only require randomization in a limited way. For our positive results for classical sequential posted price mechanisms, only the offered prices need to be randomly chosen, while the order in which the buyers are picked is arbitrary. This makes these positive results hold through for the order-oblivious (i.e., on-line) setting in which the mechanism has no control over the agent that is picked in each iteration. Our positive result for blind offer mechanisms only requires randomized pricing in case k < n and works for any ordering in which the buyers are picked, as long as the mechanisms knows the ordering in advance. Our positive result for enhanced sequential posted price mechanisms requires randomized pricing and the assumption that the mechanism can pick a uniformly random ordering of the buyers (i.e., it holds in the random order model (ROM) of arrivals). Some of the proofs have been omitted in the main body of the paper, and can be found in the appendices. 2 Preliminaries and notation For a N, we write [a] to denote the set {,..., a}. We write [X] to denote the indicator function for property X (i.e., it evaluates to if X holds, and to 0 otherwise). When v is a vector and a is an arbitrary element, we denote by (a, v i ) the vector obtained by replacing v i with a. We face a setting where an auctioneer provides a service to n buyers, and is able to serve at most k of the buyers. As mentioned in the introduction, the buyers have valuations for the service offered, which are drawn from a valuation distribution, defined as follows. Definition (Valuation distribution). A valuation distribution π for n buyers is a probability distribution on R n 0. We will assume throughout this paper that π is discrete, except for in Theorem 2, where we assume that various standard assumptions about continuous valuation distributions hold, such as regularity, the monotone hazard rate (MHR) condition and affiliation. We refer the interested reader to Appendix A for a definition and a brief discussion of these properties. We will use the following notation for conditional and marginal probability distributions. For an arbitrary probability distribution π, denote by supp(π) the support of π. Let π be a discrete finite probability distribution on R n, let i [n], S [n] and v R n. We denote by v S the vector obtained by removing from v the coordinates in [n] \ S. We denote by π S the probability distribution induced by drawing a vector from π and removing the coordinates corresponding to index set [n] \ S. If S = {i} is a singleton, we write π i instead of π {i} and if S consists of all but one buyer i, we write π i instead of π [n]\{i}. We denote by π vs the probability distribution of π conditioned on the event that v S is the vector of values on the coordinates corresponding to index set S. We denote by π i, vs the marginal probability distribution of the coordinate of π vs that corresponds to Buyer i. Again, in these cases, in the subscript we will also write i instead of {i} and i instead of [n] \ {i}. Each of the buyers is interested in receiving the service at most once. The auctioneer runs a mechanism with which the buyers interact. In general, a mechanism consists of a specification of (i) the strategies available to the buyers, (ii) a function that maps each vector of strategies chosen by the buyers to an outcome. The mechanism, when provided with a strategy profile of the buyers, outputs an outcome that consists of a vector x = (x,..., x n ) and a vector p = (p,..., p n ): vector x is the allocation vector, i.e., the (0, )-vector that indicates to which of the buyers the auctioneer allocates the service, and p = (p,..., p n ) is the vector of prices that the auctioneer asks from the buyers. For outcome ( x, p), the utility of a buyer i [n] is x i v i p i. The auctioneer is interested in maximizing the revenue i [n] p i, and is assumed to have full knowledge of the valuation distribution, but not of the actual valuations of the buyers. We formalize the above as follows. Definition 2. An instance is a triple (n, π, k), where n is the number of participating buyers, π is the valuation distribution, and k N is the number of services that the auctioneer may allocate to the buyers. A deterministic mechanism f is a function from i [n] Σ i to {0, } n R n 0, for any choice of strategy sets Σ i, i [n]. When Σ i = supp(π i ) for all 3 When the valuation distribution is not accessible in such a form, and can instead only be sampled from, then standard sampling techniques can be used in order to obtain an estimate of the distribution. When this estimate is reasonably accurate (which should be the case when the distribution in question does not have extreme outliers), then the mechanisms in our paper can still be used with only a small additional loss in revenue. 5
i [n], mechanism f is called a deterministic direct revelation mechanism. A randomized mechanism M is a probability distribution over deterministic mechanisms. For i [n] and s j [n] Σ j, we will denote i s expected allocation E f M [ f ( s) i ] by x i ( s) and i s expected payment E f M [ f ( s) n+i ] by p i ( s). (Whenever we use this notation, the mechanism M will always be clear from context.) Definition 3. Let (n, π, k) be an instance and M be a randomized direct revelation mechanism for (n, π, k). Mechanism M is dominant strategy incentive compatible (dominant strategy IC) iff for all i [n] and s j [n] supp(π j ) and v i supp(π i ), x i (v i, s i )v i p i (v i, s i ) x i ( s)v i p i ( s). Mechanism M is ex-post individually rational (ex-post IR) iff for all i [n] and s supp(π), x i (s)v i p i (s) 0. For convenience we usually will not treat a mechanism explicitly as a probability distribution over deterministic mechanisms, but rather as the result of a randomized procedure that interacts in some way with the buyers. In this case we say that a mechanism is implemented by that procedure. Sequential posted price mechanisms are the mechanisms that are implemented by a particular such procedure, defined as follows. Definition 4. An sequential posted price mechanism for an instance (n, π, k) is any mechanism that is implementable by iteratively selecting a buyer i [n] that has not been selected in a previous iteration, and proposing a price p i for the service, which the buyer may accept or reject. If i accepts, he gets the service and pays p i, resulting in a utility of v i p i for i. If i rejects, he pays nothing and does not get the service, resulting in a utility of 0 for i. Once the number of buyers that have accepted an offer equals k, the process terminates. Randomization in the selection of the buyers and prices is allowed. We will initially be concerned with only sequential posted price mechanisms. Later in the paper we define and study the two generalizations of sequential posted price mechanisms that we mentioned in the introduction. Note that each buyer in a sequential posted price mechanism has an obvious dominant strategy: He will accept whenever the price offered to him does not exceed his valuation, and he will reject otherwise. Also, a buyer always ends up with a non-negative utility when participating in a sequential posted price mechanism. Thus, by the revelation principle (see, e.g., [Börgers, 205]), a sequential posted price mechanism can be straightforwardly converted into a dominant strategy IC and ex-post IR direct revelation mechanism that achieves the same expected revenue. Our interest lies in analyzing the revenue performance of sequential posted price mechanisms. We do this by comparing the expected revenue of such mechanisms to the maximum expected revenue that can be obtained by a mechanism that satisfies dominant strategy IC and ex-post IR. Thus, for a given instance let OPT be the maximum expected revenue that can be attained by a dominant strategy IC, ex-post IR mechanism and let REV(C) be the maximum expected revenue achievable by some class of mechanisms C. Our goal throughout this paper is to derive instance-independent lower and upper bounds on the ratio REV(C)/OPT, when C is the class of sequential posted price mechanisms or one of the generalizations mentioned. A more general class of mechanisms is formed by the ex-post incentive compatible, ex-post individually rational mechanisms. Definition 5. Let (n, π, k) be an instance and M be a randomized direct revelation mechanism for (n, π, k). Mechanism M is ex-post incentive compatible (ex-post IC) iff for all i [n], s i supp(π i ) and v supp(π), x i ( v)v i p i ( v) x i (s i, v i )v i p i (s i, v i ). In other words, a mechanism is ex-post IC if it is always (i.e., for any valuation vector) a pure equilibrium for the buyers to report their valuation. In this work we sometimes compare the expected revenue of our (dominant strategy IC and ex-post IR) mechanisms to the maximum expected revenue of the more general class of ex-post IC, ex-post IR mechanisms. This strengthens our positive results. We refer the interested reader to Roughgarden and Talgam-Cohen [203] for a further discussion of and comparison between various notions of incentive compatibility and individual rationality. 6
3 Sequential posted price mechanisms We are interested in designing a posted price mechanism that, for any given n and valuation distribution π, achieves an expected revenue that lies only a constant factor away from the optimal expected revenue that can be achieved by a dominant strategy IC, ex-post IR mechanism. In this section we show that this is unfortunately impossible. In fact, we will show that the non-constant approximation ratios established for the examples in this section are asymptotically optimal in terms of a distributional parameter to be defined in Section 3.2. 3. Non-existence of good posted price mechanisms We next prove the following theorem. Theorem. For all n N 2, there exists a valuation distribution π such that for all k [n] there does not exist a sequential posted price mechanism for instance (n, π, k) that extracts a constant fraction of the expected revenue of the optimal dominant strategy IC, ex-post IR mechanism. Proof. We first consider the unit supply setting, i.e., instances of the form (n, π, ). As a first step, we show that a posted price mechanism cannot achieve a constant factor approximation of the expected optimal social welfare, defined as: OSW = E v π [max{v i : i [n]}]. Let OR be the optimal revenue that a dominant strategy IC and ex-post IR mechanism can achieve on (n, π, k). (OR depends on the valuation distribution π, but we assume that the valuation distribution is given, and implicit from context.) It is clear that OSW is an upper bound to OR regardless of π, since a dominant strategy IC and ex-post IR mechanism will not charge (in expectation) any buyer a higher price than its expected valuation. Fix m N arbitrarily, and consider the case where n = and the valuation v of the single buyer is taken from {/a: a [m]} distributed such that π (/a) = /m for all a [m]. In this setting, a posted price mechanism will offer the buyer a price p, which the buyer subsequently accepts iff v p. After that, the mechanism terminates. Note that OSW = (/m) m a= (/a). The expected revenue of the mechanism is Therefore: {a: /a p} RM = ppr v π [v p] = p = {a: /a /p } = p m mp mp = m. () lim m RM OSW = lim m a [m] /a = lim m H(m) = 0, where H(m) denotes the m-th harmonic number. So, no posted price mechanism can secure in expectation a revenue that lies a constant factor away from the expected optimal social welfare. (Because our analysis is for an instance with only one buyer, this inapproximability result also holds for instances with independent valuations.) We extend the above example in a simple way to a setting where the expected revenue of the optimal dominant strategy IC, ex-post IR mechanism is equal to the expected optimal social welfare. Fix m N and consider a setting with 2 buyers, where the type vector (v, v 2 ) takes values in {(/a, /a): a [m]} according to the probability distribution where π((/a, /a)) = /m for all a [m]. A mechanism that always gives Buyer the service, and charges Buyer the bid of Buyer 2, is clearly dominant strategy IC and also clearly achieves a revenue equal to the optimal social welfare. In this two-buyer setting, the value OSW is again (/m) m a= (/a). By symmetry, we may assume without loss of generality that a posted price mechanism works by first proposing a price p to Buyer, and then proposing a price p 2 to Buyer 2 if Buyer rejected the offer. Using arguments similar to (), we derive that the revenue of this mechanism is: RM = p Pr (v,v 2 ) π[v p ] + p 2 Pr (v,v 2 ) π[v < p v 2 p 2 ] = m + p 2Pr (v,v 2 ) π[v < p v 2 p 2 ] m + p 2Pr (v,v 2 ) π[v 2 p 2 ] = 2 m. Therefore: RM lim m OR = lim m RM OSW lim m 2 a [m] /a = 0. 7
The above example establishes the non-existence of a good sequential posted price mechanism in the case where the service has to be provided to a single buyer. Suppose now that the service can be provided to two buyers (i.e., k = 2), and each buyer gets the service at most once. Consider again two buyers whose values are drawn from the probability distribution π as defined above. As above, by symmetry we may assume that our posted price mechanism first proposes price p to Buyer, and then proposes either price p 2 or p 2 to Buyer 2: p 2 is proposed in case the offer was rejected by Buyer, and p 2 is proposed otherwise. The difference with the previous analysis for the unit supply case is that the mechanism proposes a price to Buyer 2 regardless of whether Buyer accepted the offer or not. We derive: RM = p Pr (v,v 2 ) π[v p v 2 < p 2 ] + p 2Pr (v,v 2 ) π[v < p v 2 p 2 ] + (p + p 2 )Pr (v,v 2 ) π[v p v 2 p 2 ] 2 m + (p + p 2 )Pr (v,v 2 ) π[v p v 2 p 2 ] 2 m + 2 max{p, p 2 }Pr (v,v 2 ) π[v max{p, p 2 }] 4 m. The optimal incentive compatible mechanism works by giving the service to both buyers while charging the bid of Buyer to Buyer 2, and charging the bid of Buyer 2 to Buyer. The resulting expected revenue is exactly the expected optimal social welfare: OR = OSW = (/m) m a= (2/a). We therefore obtain RM lim m OR = lim m RM OSW lim m 4 a [m] 2/a = 0. The above yields an impossibility result for 2-limited supply. By adding to this instance dummy buyers that always have valuation 0, we obtain an impossibility result for k-limited supply, where k N. We prove that the above impossibility result holds also in the continuous case even if we assume that all of the following conditions simultaneously hold: the valuation distribution is supported everywhere, is entirely symmetric, satisfies regularity, satisfies the monotone hazard rate condition, satisfies affiliation, all the induced marginal distributions have finite expectation, and all the conditional marginal distributions are non-zero everywhere. We remark that Roughgarden and Talgam-Cohen [203] showed that when all these assumptions are simultaneously satisfied, the optimal ex-post IC and ex-post IR mechanism is the Myerson Mechanism, that is the same mechanism that is optimal in the independent value setting. Thus, these conditions make the correlated setting very similar to the independent one with respect to revenue maximization. Yet, our following result shows that, whereas posted price mechanisms can achieve a constant approximation revenue in the independent setting, this result does not extend to the correlated setting. Theorem 2. There exists a valuation distribution π with the properties that. π has support [0, ] 2 ; 2. the expectation E v π [v i ] is finite for any i {, 2}; 3. π is symmetric in all its arguments; 4. π is continuous and nowhere zero on [0, ] 2 ; 5. the conditional marginal densities π i v i are nowhere zero for any v i [0, ] n and any i {, 2}; 6. π has a monotone hazard rate and is regular; 7. π satisfies affiliation; such that there does not exist a sequential posted price mechanism for instance (2, π, k), for k {, 2}, that extracts a constant fraction of the expected revenue of the optimal dominant strategy IC, ex-post IR mechanism. 8
Proof. Consider c > and m (c log c)/2 and set M = + /m. Let V be a random variable whose value is drawn over the support [/m, ] according to the probability density function f V (x) = (m )x 2. Let N and N 2 be two random variables whose values are independently drawn over the support [0, M v] according to the conditional density function f N V (z V = v) = c z ln(c) Z(v) with c > and Z(v) = c (M v). Finally, let f be the probability density function of the pair (X, Y) = (V + N /m, V + N 2 /m). Properties, 2 and 3 are trivial and can be immediately checked. For v [0, ], let f X V ( V = v) and f Y V ( V = v) be respectively the probability density functions of X and Y conditioned on the event that V = v. In order to establish the remaining properties, observe that f X V (x V = v) = f N V (x v + /m V = v) if x + /m v and 0 otherwise. Equivalently, f Y V (y V = v) = f N V (y v + /m V = v) if y + /m v and 0 otherwise. Consider now the triple (X, Y, V). The joint density function of this triple is f X,Y,V (x, y, v) = f X Y,V (x Y = y, V = v) f Y V (y V = v) f V (v). Note that f X Y,V (x Y = y, V = v) = f X V (x V = v) if min{x, y} + /m v and 0 otherwise. Then f X,Y,V (x, y, v) = f N V (x v + /m V = v) f N V (y v + /m V = v) f V (v). if min{x, y} + /m v and 0 otherwise. Hence, we can compute f as follows: f (x, y) = /m f X,Y,V (x, y, v)dv = ln2 (c) α+/m m c 2v c (x+y) /m v 2 Z(v) dv, 2 where α = min { /m, x, y}. Note that the integrated function is continuous and positive in the interval in which it is integrated. Hence, the integral turns out to be non-zero. From this, we observe that f (x, y) is continuous and nowhere zero on [0, ] 2, satisfying Property 4. Let us now derive the conditional probability density functions. By symmetry it will be sufficient to focus only on f X Y. f X Y (x Y = y) = f X Y,V (x Y = y, V = v) f V (v)dv = ln(c) /m m c x = m2 c M ln(c) α c x m 0 (mz + ) 2 (c z ) dz, α+/m /m c v v 2 Z(v) dv where the last equality is obtained by setting z = v m. It is now obvious that the conditional probability density functions are continuous and nowhere zero, as desired by Property 5. Let γ(z) = /((mz + ) 2 (c z )), g(a) = a γ(z)dz with a [0, ] and let 0 α = min{y, /m}. Then Moreover, we have that f X Y (x Y = y) = m2 c M ln(c) m c x g(x), if x < α ; g(α ), otherwise. F X Y (x Y = y) = f X Y (z Y = y)dz x = m2 c M α ln(c) c z g(z)dz + g(α ) c z dz, if x < α ; x α m g(α ) x c z dz = g(α )(c x c ) ln(c), otherwise. 9
Hence, the inverse hazard rate is I(x) = F X Y(x Y = y) f X Y (x Y = y) = α x c z g(z)dz+g(α ) α c z dz c x ln(c), c x g(x), if x < α ; otherwise. We prove that I(x) is non-increasing in x in the interval [0, ] and thus f has the monotone hazard rate and is, as a consequence, regular, as required by Property 6. Clearly, I(x) is non-increasing in x in the interval [α, ] since in this case I(x) = ( c x )/ ln(c). Moreover, I(x) does not have discontinuities for x = α. So, it is sufficient to show that I(x) in non-increasing also in the interval [0, α ]. To this aim, observe that for x < α, di(x) dx α = d c z g(z)dz + g(α ) c z dz x α dx c x g(x) ( = c x g(x) d α α c z g(z)dz dx +c x g(x)g(α ) d dx x α c z dz g(α ) x c z g(z)dz d dx c x g(x) Observe that, according to the second fundamental theorem of calculus, whereas and d dx α c z dz = 0. Then, di(x) dx ( = + ln(c) γ(x) ) α x g(x) d α c z g(z)dz = c x g(x), dx x d dx c x g(x) = c x γ(x) c x g(x) ln(c), α c z dz d ) dx c x g(x) /(c x g(x)) 2. c z g(z)dz + g(α ) c z ( dz α c x = + ln(c) γ(x) ) I(x). g(x) g(x) The result then follows by showing that γ(x)/g(x) ln(c). To this aim, let us consider the function γ (z) = c z (c z )(mz + ) 2 for z [0, x]. Note that dγ ( ) ( ) (z) 2m + mz log c + log c 2m + log c = 2mc(mz + ) 2mc(mz + ) 0, dz c z c where we used the fact that m (c log c)/2. Thus, γ (z) is non-increasing in its argument and, in particular, γ (z) γ (x) (c x )(c x )(mx + ) 2. By simple algebraic manipulation, it then follows that γ(z) γ(x)c z /(c x ). Then as desired. Set now C = ln 2 (c)/(m ) and let γ(x) g(x) = γ(x) x γ(z)dz γ(x) x c z 0 0 c x γ(x)dz = cx x 0 cz dz = ln(c), h(a) = min{ /m,a}+/m /m c 2v v 2 Z(v) 2 dv with a [0, ]. Note that integrated function is positive for any v [/m, ]. Hence, the integral increases as the size of the interval in which it is defined increases. In other words, the function h(a) is non-decreasing in a. 0
Consider now two pairs (x, y) and (x, y ). Moreover, let ˆx = max{x, x } and ˇx = min{x, x } and, similarly, define ŷ and ˇy. We show that f (x, y) f (x, y ) f ( ˆx, ŷ) f ( ˇx, ˇy), satisfying in this way also Property 7. Indeed, f (x, y) f (x, y ) = C 2 c (x+y+x +y ) h(min{x, y})h(min{x, y }). If x x and y y (x < x and y < y, respectively), then ( ˆx, ŷ) = (x, y) ((x, y ), resp.) and ( ˇx, ˇy) = (x, y ) ((x, y), resp.), and the desired result immediately follows. Suppose instead that ( ˆx, ŷ) = (x, y ) and ( ˇx, ˇy) = (x, y). Then f ( ˆx, ŷ) f ( ˇx, ˇy) = C 2 c (x+y+x +y ) h(min{x, y })h(min{x, y}). We will prove that in this case h(min{x, y})h(min{x, y }) h(min{x, y })h(min{x, y}). First observe that on both sides one of the two factors must be h(min{x, y, x, y }) = h(min{x, y}). Suppose without loss of generality, that min{x, y} = y. Then it is sufficient to prove that h(min{x, y }) h(min{x, y }), or, since h in non-decreasing, that min{x, y } min{x, y }. If x y, then x x y by hypothesis and the claim follows. If y < x, then it immediately follows that min{x, y } y. The case that ( ˆx, ŷ) = (x, y) and ( ˇx, ˇy) = (x, y ) is similar and hence omitted. Finally, observe that lim c f N V (0 V = v) = and lim c f N V (z V = v) = 0 for any z > 0. Hence, lim c X = lim c Y = V /m. Let us consider the case that the service can be offered to only one buyer. In this setting, the following is a dominant strategy IC and ex-post IR mechanism: it offers to Buyer the service at a price equivalent to the valuation of Buyer 2 minus a fixed constant ɛ. For small enough c, ɛ can be chosen arbitrarily small. Thus, for any ɛ there exists a choice of c such that in expectation this mechanism extracts as revenue all but ɛ of the social welfare. The expected optimal social welfare (and thus the optimal expected revenue) is: OR = OSW = E v fv [v /m] = /m v /m ln(m) dv = (m )v2 m ln(m ) m m. A posted price mechanism will offer Buyer a price p 0, which the buyer subsequently accepts iff X p. After that, if Buyer rejects, the mechanism offers a price p 2 to Buyer 2. Thus, if p [0, /m], then p Pr v fv [X p ] = p Pr v fv [v p + /m] = p p +/m (m )v 2 dv = p m Moreover, if p /m, then p Pr v f [X p ] = 0. Hence, ( ) p + /m p m m. Therefore: RM = p Pr v f [v p ] + p 2 Pr v f [v < p v p 2 ] RM lim m OR = lim m RM OSW lim m m + p 2 2Pr v f [v p 2 ] m. 2 ln(m ) = 0. The case in which the service can be offered to both buyer is similar and omitted. We note that above theorem can be easily extended to an arbitrary number of buyers, by adding dummy buyers whose valuation is independently drawn over the support [0, ] according to the probability density function f (x) = c x ln c c. Note that, with this extension, the resulting distribution π does not satisfy the symmetry condition anymore, but on the other hand, the symmetry property is not necessary for the optimality of the Myerson mechanism to hold (see Roughgarden and Talgam-Cohen [203]).
3.2 A revenue guarantee for sequential posted price mechanisms In the previous section we have demonstrated that it is impossible to have a sequential posted price mechanism extract a constant fraction of the optimal revenue. More precisely, in our example instances it was the case that the expected revenue extracted by every posted price mechanism is a Θ(/ log(r)) fraction of the optimal expected revenue, where r is the ratio between the highest valuation and the lowest valuation in the support of the valuation distribution. A natural question that arises is whether this is the worst possible instance in terms of revenue extracted, as a function of r. We show here that this is indeed the case, asymptotically: For every valuation distribution π, there exists a mechanism that extracts in expectation at least a Θ(/ log(r)) fraction of the revenue of the optimal revenue. We note that in many realistic scenarios, we do not expect the extremal valuations of the buyers to lie too far from each other, because often the valuation of a buyer is strongly impacted by prior objective knowledge of the value of the service to be auctioned. The results of this section are valuable when that is the case. We start with the unit supply case. Definition 6. For a valuation distribution π on R n, let v max π min{max{v i : i [n]}: v supp(π)} respectively. Let r π = v max π coordinate-wise maximum valuation in the support of π. and v min π /v min π be max{v i : v supp(π), i [n]} and be the ratio between the highest and lowest Proposition. Let n N, and let π be a probability distribution on R n. For the unit supply case there exists a sequential posted price mechanism that, when run on instance (n, π, ), extracts in expectation at least an Ω(/ log(r π )) fraction of the optimal social welfare (and therefore also of the expected revenue of the optimal dominant strategy IC and ex-post IR auction). Proof. The proof uses a standard bucketing trick (see, e.g., Babaioff et al. [2007]). Let M be the sequential posted price mechanism that draws a value p uniformly at random from the set S = {v min π 2 k : k [ log(r π ) ] {0}}. M offers price p to all the bidders in an arbitrary order, until a bidder accepts. Let π max be the probability distribution of the coordinate-wise maximum of π. Note that S does not exceed log(rπ max ). Therefore the probability that p is the highest possible value (among the values in S ) that does not exceed the value drawn from π max, is equal to / log(r π ). More formally, let π S be the probability distribution from which p is drawn; then Pr v max π max,p π S [p v max ( p S : p > p p v max )] log(r π ). Thus, with probability / log(r π ), the mechanism generates a revenue of exactly v min π 2 k, where k is the number such that the value drawn from π max lies in between v min π 2 k and v min π 2 k+. This implies that with probability / log(r π ) the mechanism generates a revenue that lies a factor of at most /2 away from the optimal social welfare OPT( v) (i.e., the coordinate-wise maximum valuation): E v π [revenue of M( v)] log(r π ) 2 E v π[opt( v)] 2 log(r π ) E v π[opt( v)]. This result can be generalized to yield revenue bounds for the case of k-limited supply, where k >. We prove a more general variant of the above in Appendix C. The above result does not always guarantee a good revenue; for example in the extreme case where v min π = 0. However, it is straightforward to generalize the above theorem such that it becomes useful for a much bigger family of probability distributions: let ˆv and ˇv be two any two values in the support of π max, and let c(ˆv, ˇv) = Pr v max π max [ˇv v max ˆv]. Then by replacing the values v max π and v min π in the above proof by respectively ˆv and ˇv, we obtain a sequential posted price mechanism that extracts in expectation a c(ˆv, ˇv)/(2 log(ˆv/ˇv)) fraction of the optimal social welfare. By choosing ˆv and ˇv such that this ratio is maximized, we obtain a mechanism that extracts a significant fraction of the optimal social welfare in any setting where the valuation distribution of a buyer is concentrated in a relatively not too large interval. A better result can be given for the unlimited supply case. Definition 7. For a valuation distribution π on R n and any i [n], let v max and v min be max{v i : v supp(π)} and min{v i : v supp(π)} respectively. Let r = v max /v min, be the ratio between the highest and lowest valuation of Buyer i in the support of π. Proposition 2. Let n N, and let π be a probability distribution on R n. There exists a sequential posted price mechanism that, when run on instance (n, π, n), extracts in expectation at least an Ω(/ log(max{r : i [n]})) fraction 2
of the expected revenue of the expected optimal social welfare (and therefore also the expected revenue of the optimal dominant strategy IC and ex-post IR mechanism). The proof (found in Appendix B) applies similar techniques as those in the proof of Proposition. The techniques used for proving these results can be applied to improve the approximation guarantees for the more general k-limited supply setting, for any k [n], under special conditions. We give an example of such a result in Appendix C. Clearly, the stated bound of O(/ log(max{r : i [n]})) is very crude. For most practical settings we expect that it is possible to do a much sharper revenue analysis of the mechanisms in the proofs of the above propositions, by taking the particular valuation distribution into account. Moreover, as suggested above, also for the unlimited supply case it is possible to tweak the proof in a straightforward way in order to achieve a good revenue in cases where the ratios r are very large. Finally, note that the mechanisms in the proofs of these two propositions do not take into account any dependence and correlation among the valuations of the buyers. When provided with a particular valuation distribution, a better revenue and sharper analysis may be obtained by taking such dependence into account, and adapting the mechanisms accordingly. 4 Enhanced sequential posted price mechanisms Our negative results on sequential posted price mechanisms suggest that it is necessary for a mechanism to have a means to retrieve the valuations of some of the buyers in order to improve the revenue performance. We accordingly propose two generalizations of sequential posted price mechanisms, in such a way that they possess the ability to retrieve the valuations of some of the buyers. Definition 8. An enhanced sequential posted price mechanism for an instance (n, π, k) is a mechanism that can be implemented by iteratively selecting a buyer i [n] that has not been selected in a previous iteration, and performing exactly one of the following actions on Buyer i: Offer the service at price p i to Buyer i, which the buyer may accept or reject. If i accepts, he gets the service and pays p i, resulting in a utility of v i p i for i. If i rejects, he pays nothing and does not get the service, resulting in a utility of 0 for i. Ask i for his valuation (in which case the buyer pays nothing and does not get the service). Randomization is allowed. This generalization is still dominant strategy IC and ex-post IR (that is, the revelation principle allows us to convert them to dominant strategy IC, ex-post IR direct revelation mechanisms). Actually, one can see that this mechanism still enjoys also the stronger property of being obviously strategy-proof [Li, 205]. Indeed, for enhanced sequential posted price mechanisms, when a buyer gets asked his valuation, he has no incentive to lie, because in this case he does not get the service and he pays nothing 4. Next, we analyze the revenue performance of enhanced sequential posted price mechanisms. For this class of mechanisms we prove that it is unfortunately still the case that no constant fraction of the optimal revenue can be extracted: The next section establishes an O(/n) bound for enhanced sequential posted price mechanisms. Enhanced sequential posted price mechanisms nonetheless turn out to be more powerful than standard sequential posted price mechanisms. Contrary to what we had for the former ones, enhanced mechanisms can be shown to extract a fraction of the optimal revenue that is independent of the valuation distribution: The O(/n) bound turns out to be asymptotically tight. Our main positive result for enhanced sequential posted price mechanisms is that when dependence of the valuation among the buyers is limited, then a constant fraction of the optimal revenue can be extracted. Specifically, in Section 4.3 we define the concept of d-dimensional dependence and prove that for an instance (n, π, k) where π is d-dimensionally dependent, there exists an enhanced sequential posted price mechanism that extracts an Ω(/d) fraction of the optimal revenue. (This implies the claimed Ω(/n) bound by taking d = (n ).) It is instructive to identify the basic reason(s) why, in the case of general correlated distributions, standard and enhanced sequential posted price mechanisms fail to achieve a constant approximation of the optimum revenue. There are two main limitations of (enhanced) sequential posted price mechanisms; namely, (i) such mechanisms do not 4 A problematic aspect from a practical point of view is that while there is no incentive to a buyer to lie, there is also no incentive to tell the truth. This problem is addressed in Appendix D. 3